Riven has been OP since release
Riven was regarded as UP on her release.
She also didin't see an explosion of play until fairly recently (most notably when FAKER played her in mid-lane), and even with the occasional solid pro-game: she hasn't exactly been dominating the scene.
Riven has been stomping scrubs and making casuals salty, which isn't OP - it's just harsh to new players. And with how different the landscape is at the pro scene compared to yoloqueue, there's always going to be someone harsh to newer players.
Why just make her use mana or something?

Radical shift in her itemization. Riven's always been intended as a melee-range auto-attacker, not a multi Lich-Bane chain nuker.
As for "or something"
Always saw Riven's Q as more of an AA-modifier akin to those on
than anything else.
Which is heavily supported by Q's bases and scalings - at rank 1 Riven's level-1 AAs actually do more damage, and with 225 total AD (only ~115 bonus on level 18 bases), Riven's rank 5 Wings do less damage than an AA apiece.
With AA's being manaless in the first place - I'm not sure I'd really take issue with a string of 3 auto-attacks in a sudo-AS/MS steroid that can't critically strike being manaless.
Or make her shield an actual decent cooldown?
Cooldown is fairly long and the duration is fairly short.
It's also worth pointing out that, rather than an HP shield, most light-fighters have something with 100% damage block potential.
100% evasion, refusal to die, free GA, spell-shield, invulnerability, invulnerability and immunity to CC, projectile-blocking shields, invisibility, untargetability, clones capable of eating champ-only ultimates.

Given all the alternatives (and the fact Riven's HP/5 is among the worst in the game), taking issuse with her shield is exactly backed by anything objective or tangible.
Q_Qing about Riven's shield is 100% based on saltiness.
If she gets one kill in lane she can snowball uncontrollably and 1v5 carry a team on her back. Riot says they fixed snowball champs like this a while ago. Why not fix riven???
can is very different than does, and for Riven to snowball uncontrollably, or any hyper-carry for that matter, it's more a minimum 3 kills in the first 10 minutes.
If her opponent has the same battle-strategy as Goku, Ichigo, Naruto or any other Anime hero of getting hit a whole bunch, never giving up, and trying to muscle there way through an opponent's strength via sheer willpower: then yes - 1 kill probably will quickly turn into 3 for Riven.
But not every top-laner is a complete and total moron.
You can certainly recover from 1 death to Riven. You can recover from 2 with some solid effort. With 3 - you generally need some fairly coordinated team-work and solid communication, exploiting Riven's fairly lacking team-fight presence with forcing 5v5s and preventing her from shinning in 1v1 scenarios.
All of this kind of assumes everyone's not completely bloodthirsty and suicidal where +3 kills Riven is 9/6 and her lane opponent is still 6/9: which is a significantly better spot than being 0/3 vs. a 3/0 champion.
1 kill in lane is no guarantee on a snowball though, and it isn't a snowball in and of itself on Riven.